Fractional Power Series Solutions for Systems of Equations

Abstract

In this paper we extend the explorations in [8] to include the fractional power series expansions of k equations in d variables, where d>k . An analog of Newton's polygon construction which uses the Minkowski sum P of the Newton polytopes P 1 ,...,P k of the k equations is given for computing such series expansions. If the Newton polytopes of these equations are the same, then the common domains of convergence for the solutions correspond to the vertices of a certain fiber polytope Σ(P) . In general, our results suggest the existence of a ``mixed fiber polytope'' of k polytopes. It is also indicated that there may be a relationship between these mixed fiber polytopes and a generalization of the discriminant, which we call the mixed discriminant.